The Theory of Pseudo-differential Operators on the Noncommutative n-Torus

TL;DR

This paper rigorously develops the pseudo-differential calculus on noncommutative n-tori, filling in technical details, defining Sobolev spaces, and extending results to modules, thereby advancing the mathematical foundation of noncommutative geometry.

Contribution

It provides detailed proofs and extends the pseudo-differential calculus to noncommutative tori and their modules, strengthening the analytical framework in noncommutative geometry.

Findings

Reproved formulas for symbols of adjoint and product of pseudo-differential operators.

Defined Sobolev spaces on noncommutative tori and proved related lemmas.

Extended results to finitely generated projective modules over noncommutative tori.

Abstract

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry and in these contexts require extensive use of pseudo-differential operators. In a foundational paper, Connes showed that, by direct analogy with the theory of pseudo-differential operators on $\mathbb R^n$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb T_{\theta}^n$, and with the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper in which the Gauss--Bonnet theorem on $\mathbb T_{\theta}^2$ is proved and later papers. Certain details of the proofs in the original derivation of the calculus were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analog of Sobolev spaces for which we prove the Sobolev and Rellich lemmas. We then extend these results to finitely generated projective right modules over the noncommutative $n$ torus.

Equations

$$\sigma (P^{\ast })\sim \sum _{\ell \in {\mathbb{Z}}_{\ge 0}^n}\frac{{\partial }^{\ell }{\delta }^{\ell }[(\rho (\xi ){)}^{\ast }]}{{\ell }_1!{\ell }_2!}.$$

$$\sigma (QP)\sim \sum _{\ell \in {\mathbb{Z}}_{\ge 0}^n}\frac{1}{{\ell }_1!{\ell }_2!}{\partial }^{\ell }\varphi (\xi ){\delta }^{\ell }\rho (\xi ),$$

$$\sigma (P^{\ast })\sim \sum _{({\ell }_1,\dots ,{\ell }_n)\in ({\mathbb{Z}}_{\ge 0}{)}^n}\frac{{\partial }_1^{{\ell }_1}\cdots {\partial }_n^{{\ell }_n}{\delta }_1^{{\ell }_1}\cdots {\delta }_n^{{\ell }_n}(\rho (\xi ){)}^{\ast }}{{\ell }_1!\cdots {\ell }_n!}.$$

$$\sigma (PQ)\sim \sum _{({\ell }_1,\dots ,{\ell }_n)\in ({\mathbb{Z}}_{\ge 0}{)}^n}\frac{{\partial }_1^{{\ell }_1}\cdots {\partial }_n^{{\ell }_n}(\rho (\xi )){\delta }_1^{{\ell }_1}\cdots {\delta }_n^{{\ell }_n}({\rho }^{\prime }(\xi ))}{{\ell }_1!\cdots {\ell }_n!}.$$

$$\underset{m\in {\mathbb{Z}}^n}{\sup }(\prod _j|m_j{|}^{{\alpha }_j}|a_m|)<\infty .$$

$$P_{\rho }(a):=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot \xi }\rho (\xi ){\alpha }_s(a)\mathrm{d}s\mathrm{d}\xi .$$

$$\underset{ϵ\to 0}{\lim }\int e^{iq(x)}a(x)\phi (ϵx)\mathrm{d}x$$

$$\left|\int e^{iq(x)}a(x)\mathrm{d}x\right|\le C_{q,m}\underset{|\alpha |\le m+n+1}{\max }\inf \{U\in \mathbb{R}:|(1+|x{|}^2{)}^{-m/2}{\partial }^{\alpha }a|\le U\text{ almost everywhere}\}$$

$$||{\delta }_1^{i_1}\cdots {\delta }_n^{i_n}({\partial }_1^{j_1}\cdots {\partial }_n^{j_n}\rho )(\xi )||\le C_{\rho }(1+|\xi |{)}^{m-|j|}.$$

$$\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-iy\cdot \eta }a(y)\mathrm{d}y\mathrm{d}\eta =\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-iy\cdot \eta }a(\eta )\mathrm{d}y\mathrm{d}\eta =a(0).$$

$$P_{\rho }(a)=\sum _{m\in {\mathbb{Z}}^n}\rho (m)a_m\prod _{j=1}^n{U}_j^{m_j}.$$

$$P_{\rho }\left(\prod _{j=1}^n{U}_j^{m_j}\right)$$

$$=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot \xi }\rho (\xi )e^{is\cdot m}\prod _{j=1}^n{U}_j^{m_j}\mathrm{d}s\mathrm{d}\xi$$

$$=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot (\xi -m)}\rho (\xi )\mathrm{d}s\mathrm{d}\xi \prod _{j=1}^n{U}_j^{m_j}$$

$$=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot \eta }\rho (\eta +m)\mathrm{d}s\mathrm{d}\eta \prod _{j=1}^n{U}_j^{m_j}$$

$$=\rho (m)\prod _{j=1}^n{U}_j^{m_j},$$

$$P_{\rho }(a)=\sum _{m\in {\mathbb{Z}}^n}\rho (m)a_m\prod _{j=1}^n{U}_j^{m_j},$$

$$\sigma (P^{\ast })\sim \sum _{\ell \in {\mathbb{Z}}_{\ge 0}^n}\frac{{\partial }^{\ell }{\delta }^{\ell }[(\rho (\xi ){)}^{\ast }]}{{\ell }_1!{\ell }_2!}.$$

$$⟨P_{\rho }(a),b⟩$$

$$=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot \xi }\tau (b^{\ast }\rho (\xi ){\alpha }_s(a))\mathrm{d}s\mathrm{d}\xi$$

$$=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot \xi }\tau ({\alpha }_{-s}(\rho (\xi {)}^{\ast }b{)}^{\ast }a)\mathrm{d}s\mathrm{d}\xi$$

$$=\tau \left({\left(\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{+is\cdot \xi }{\alpha }_{-s}(\rho (\xi {)}^{\ast }b)\mathrm{d}s\mathrm{d}\xi \right)}^{\ast }a\right)$$

$$=⟨a,P_{\rho }^{\ast }(b)⟩$$

$$P_{\rho }^{\ast }(b)$$

$$=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{+is\cdot \xi }{\alpha }_{-s}(\rho (\xi {)}^{\ast }){\alpha }_{-s}(b)\mathrm{d}s\mathrm{d}\xi$$

$$=\sum _{m,k}\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{+is\cdot \xi }{e}^{+is\cdot m}{\rho }_m(\xi {)}^{\ast }{e}^{-is\cdot k}{b}_k\mathrm{d}s\mathrm{d}\xi U_n^{-m_n}\cdots U_1^{-m_1}{U}_1^{k_1}\cdots U_n^{k_n}$$

$$=\sum _{m,k}\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-is\cdot \eta }{\rho }_m((k-m)-\eta {)}^{\ast }{b}_k\mathrm{d}s\mathrm{d}\eta U_n^{-m_n}\cdots U_1^{-m_1}{U}_1^{k_1}\cdots U_n^{k_n}$$

$$=\sum _{m,k}{\rho }_m(k-m{)}^{\ast }{b}_k{U}_n^{-m_n}\cdots U_1^{-m_1}{U}_1^{k_1}\cdots U_n^{k_n}$$

$$=\sum _{m,k}({\rho }_m(k-m)U_1^{m_1}\cdots U_n^{m_n}{)}^{\ast }(b_k{U}_1^{k_1}\cdots U_n^{k_n}),$$

$$\sigma (P_{\rho }^{\ast })(\xi )$$

$$={\left[\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-ix\cdot y}\sum _m{\rho }_m(\xi -y){\alpha }_x\left(\prod _{j=1}^n{U}_j^{m_j}\right)\mathrm{d}x\mathrm{d}y\right]}^{\ast }$$

$$={\left[\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-ix\cdot y}{\alpha }_x(\rho (\xi -y))\mathrm{d}x\mathrm{d}y\right]}^{\ast }$$

$$\rho (\xi -y)$$

$${\alpha }_x(\rho (\xi -y))$$

$$R_{N_1}(\xi ,y)=N_1\sum _{|\ell |=N_1}\frac{(-y{)}^{\ell }}{\ell !}{\int }_0^1(1-\gamma {)}^{N_1-1}({\partial }^{\ell }\rho )(\xi -y\gamma )\mathrm{d}\gamma$$

$$\sigma (P_{\rho }^{\ast })(\xi )$$

$$=\sum _{|\ell |<N_1}\frac{{\partial }^{\ell }{\delta }^{\ell }[(\rho (\xi ){)}^{\ast }]}{\ell !}+T_{N_1}(\xi ,y{)}^{\ast }$$

$$T_{N_1}(\xi ,y)=\frac{1}{(2\pi {)}^n}{\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}{e}^{-ix\cdot y}{\alpha }_x(R_{N_1}(\xi ,y))\mathrm{d}x\mathrm{d}y$$

$$\sum _{N\le \ell <N_1}\frac{{\partial }^{\ell }{\delta }^{\ell }[(\rho (\xi ){)}^{\ast }]}{\ell !}\in S^{M-N},$$

Full text

The Theory of Pseudo-differential Operators

on the Noncommutative $n$-Torus

Jim Tao

1 Introduction

The methods of spectral geometry are useful for investigating the metric aspects of noncommutative geometry [3, 4, 1, 5] and in these contexts require extensive use of pseudo-differential operators. In the foundational paper [2], Connes showed that, by direct analogy with the theory [12, 13, 14] of pseudo-differential operators on $\mathbb{R}^{n}$, one may derive a similar pseudo-differential calculus on noncommutative $n$ tori $\mathbb{T}_{\theta}^{n}$. With the development of this calculus came many results concerning the local differential geometry of noncommutative tori for $n=2,4$, as shown in the groundbreaking paper [7] in which the Gauss–Bonnet theorem on $\mathbb{T}_{\theta}^{2}$ is proved and later papers [6, 8, 9, 11, 10]. In these papers, the flat geometry of $\mathbb{T}_{\theta}^{n}$ which was studied in [2] is conformally perturbed using a Weyl factor given by a positive invertible smooth element in $C^{\infty}(\mathbb{T}_{\theta}^{n})$. Connes’ pseudodifferential calculus is critically used to apply heat kernel techniques to geometric operators on $\mathbb{T}_{\theta}^{n}$ to derive small time heat kernel expansions that encode local geometric information such as scalar curvature. As discovered in [6, 9, 11], a purely noncommutative feature that appears in the computations and in the final formula for the curvature is the modular automorphism of the state implementing the conformal perturbation of the metric.

Certain details of the proofs in the derivation of the calculus in [2] were omitted, such as the evaluation of oscillatory integrals, so we make it the objective of this paper to fill in all the details. After reproving in more detail the formula for the symbol of the adjoint of a pseudo-differential operator and the formula for the symbol of a product of two pseudo-differential operators, we define the corresponding analog of Sobolev spaces for which we prove the Sobolev and Rellich lemmas. We then extend these results to finitely generated projective right modules over the noncommutative $n$ torus.

We list these results below.

Theorem 1.1**.**

Suppose $P$ is a pseudodifferential operator with symbol $\sigma(P)=\rho=\rho(\xi)$ of order $M$. Then the symbol of the adjoint $P^{*}$ is of order $M$ and satisfies

[TABLE]

Theorem 1.2**.**

Suppose that $P$ is a pseudodifferential operator with symbol $\sigma(P)=\rho=\rho(\xi)$ of order $M_{1}$, and $Q$ is a pseudodifferential operator with symbol $\sigma(Q)=\phi=\phi(\xi)$ of order $M_{2}$. Then the symbol of the product $QP$ is of order $M_{1}+M_{2}$ and satisfies

[TABLE]

where $\partial^{\ell}:=\prod_{j}\partial_{j}^{\ell_{j}}$ and $\delta^{\ell}:=\prod_{j}\delta_{j}^{\ell_{j}}$.

Theorem 1.3**.**

For $s>k+1$, $H^{s}\subseteq A_{\theta}^{k}$.

Theorem 1.4**.**

Let $\{a_{N}\}\in A_{\theta}^{\infty}$ be a sequence. Suppose that there is a constant $C$ so that $||a_{N}||_{s}\leq C$ for all $N$. Let $s>t$. Then there is a subsequence $\{a_{N_{j}}\}$ that converges in $H^{t}$.

Theorem 1.5**.**

1. (a)

For a pseudodifferential operator $P$ with $r\times r$ matrix valued symbol $\sigma(P)=\rho=\rho(\xi)$, the symbol of the adjoint $P^{*}$ satisfies

[TABLE] 2. (b)

If $Q$ is a pseudodifferential operator with $r\times r$ matrix valued symbol $\sigma(Q)=\rho^{\prime}=\rho^{\prime}(\xi)$, then the product $PQ$ is also a pseudodifferential operator and has symbol

[TABLE]

Theorem 1.6**.**

For $s>k+1$, $H^{s}\subseteq(A_{\theta}^{k})^{r}e$.

Theorem 1.7**.**

Let $\{\vec{a}_{N}\}\in(A_{\theta}^{\infty})^{r}e$ be a sequence. Suppose that there is a constant $C$ so that $||\vec{a}_{N}||_{s}\leq C$ for all $N$. Let $s>t$. Then there is a subsequence $\{\vec{a}_{N_{j}}\}$ that converges in $H^{t}$.

2 Preliminaries

Fix some skew symmetric $n\times n$ matrix $\theta$ with upper triangular entries in $\mathbb{R}\backslash\mathbb{Q}$ that are linearly independent over $\mathbb{Q}$. Consider the irrational rotation $C^{*}$-algebra $A_{\theta}$ with $n$ unitary generators $U_{1},\ldots,U_{n}$ which satisfy $U_{k}U_{j}=e^{2\pi i\theta_{j,k}}U_{j}U_{k}$ and $U_{j}^{*}=U_{j}^{-1}$. Let $\{\alpha_{s}\}_{s\in\mathbb{R}^{n}}$ be a $n$-parameter group of automorphisms given by $\prod_{j}U_{j}^{m_{j}}\mapsto e^{is\cdot m}\prod_{j}U_{j}^{m_{j}}$. We define the subalgebra $A_{\theta}^{k}$ of $C^{k}$ elements of $A_{\theta}$ to be those $a\in A_{\theta}$ such that the mapping $\mathbb{R}^{n}\rightarrow A_{\theta}$ given by $s\mapsto\alpha_{s}(a)$ is $C^{k}$, and we define the subalgebra $A_{\theta}^{\infty}$ of smooth elements of $A_{\theta}$ to be those $a\in A_{\theta}$ such that the mapping $\mathbb{R}^{n}\rightarrow A_{\theta}$ given by $s\mapsto\alpha_{s}(a)$ is smooth. An alternative definition of the subalgebra $A_{\theta}^{\infty}$ of smooth elements is the elements in $A_{\theta}$ that can be expressed by an expansion of the form $\sum_{m\in\mathbb{Z}^{n}}a_{m}\prod_{j}U_{j}^{m_{j}}$, where the sequence $\{a_{m}\}_{m\in\mathbb{Z}^{n}}$ is in the Schwartz space $\mathcal{S}(\mathbb{Z}^{n})$ in the sense that, for all $\alpha\in\mathbb{Z}^{n}$,

[TABLE]

Define the trace $\tau:A_{\theta}\rightarrow\mathbb{C}$ by $\tau(\prod_{j}U_{j}^{m_{j}})=0$ for $m_{j}$ not all zero and $\tau(1)=1$ and define an inner product $\langle\cdot,\cdot\rangle:A_{\theta}\times A_{\theta}\rightarrow\mathbb{C}$ by $\langle a,b\rangle=\tau(b^{*}a)$ with induced norm $||\cdot||:A_{\theta}\rightarrow\mathbb{R}_{\geq 0}$. Let $D_{j}=-i\partial_{j}$ and define derivations $\delta_{j}$ by the relations $\delta_{j}(U_{j})=U_{j}$ and $\delta_{j}(U_{k})=0$ for $j\neq k$. For convenience, denote $\partial^{\ell}:=\prod_{j}\partial_{j}^{\ell_{j}}$, $\delta^{\ell}:=\prod_{j}\delta_{j}^{\ell_{j}}$, and $D^{\ell}:=\prod_{j}D_{j}^{\ell_{j}}$. We define a map $\psi:\rho\mapsto P_{\rho}$ assigning a pseudo-differential operator on $A_{\theta}^{\infty}$ to a symbol $\rho\in C^{\infty}(\mathbb{R}^{n},A_{\theta}^{\infty})$.

Definition 2.1**.**

For $\rho\in C^{\infty}(\mathbb{R}^{n},A_{\theta}^{\infty})$, let $P_{\rho}$ be the pseudo-differential operator sending arbitrary $a\in A_{\theta}^{\infty}$ to

[TABLE]

The integral above does not converge absolutely; it is an oscillatory integral. We define oscillatory integrals below as in [13].

Definition 2.2**.**

Let $q$ be a nondegenerate real quadratic form on $\mathbb{R}^{n}$, $a$ be a $C^{\infty}$ complex-valued function defined on $\mathbb{R}^{n}$ such that the functions $(1+|x|^{2})^{-m/2}\partial^{\alpha}a(x)$ are bounded on $\mathbb{R}^{n}$ for all $\alpha\in\mathbb{Z}_{\geq 0}^{n}$, and $\varphi$ be a Schwartz function, i.e. the functions $x^{\alpha}\partial^{\beta}\varphi(x)$ are bounded on $\mathbb{R}^{n}$ for all pairs $\alpha,\beta\in\mathbb{Z}_{\geq 0}^{n}$. Suppose further that $\varphi(0)=1$. Then the limit

[TABLE]

exists, is independent of $\varphi$ (as long as $\varphi(0)=1$), and is equal to $\int\!e^{iq(x)}a(x)\,\mathrm{d}x$ when $a\in L^{1}$. When $a\not\in L^{1}$, we continue to denote this limit by $\int\!e^{iq(x)}a(x)\,\mathrm{d}x$, and have an estimate

[TABLE]

where $C_{q,m}$ depends only on the quadratic form $q$ and the order $m$.

As shown in [13], oscillatory integrals behave essentially like absolutely convergent integrals in that one can still make changes of variables, integrate by parts, differentiate under the integral sign, and interverse integral signs. Given certain conditions on $\rho$, $P_{\rho}(a)$ satisfies the conditions of the oscillatory integral, and we can evaluate $P_{\rho}(a)$.

Definition 2.3**.**

An element $\rho=\rho(\xi)=\rho(\xi_{1},\ldots,\xi_{n})$ of $C^{\infty}(\mathbb{R}^{n},A_{\theta}^{\infty})$ is a symbol of order $m$ if and only if for all non-negative integers $i_{1},\ldots,i_{n},j_{1},\ldots,j_{n}$

[TABLE]

Example 2.6(i) of [13] gives a convenient formula for evaluating the oscillatory integrals that appear in our calculation of $P_{\rho}(a)$.

Proposition 2.4**.**

Suppose that, for some $m$, $a$ is a $C^{\infty}$ complex-valued function defined on $\mathbb{R}^{n}$ such that the functions $(1+|x|^{2})^{-m/2}\partial^{\alpha}a(x)$ are bounded on $\mathbb{R}^{n}$ for all $\alpha\in\mathbb{Z}_{\geq 0}^{n}$. Then

[TABLE]

We apply Proposition 2.4 to get a basic result.

Lemma 2.5**.**

Let $a=\sum_{m\in\mathbb{Z}^{n}}a_{m}\prod_{j}U_{j}^{m_{j}}$ be an arbitrary element of $A_{\theta}^{\infty}$ and let $\rho\in C^{\infty}(\mathbb{R}^{n},A_{\theta}^{\infty})$ be a symbol of order $M$. Then

[TABLE]

Proof.

First consider the case $a=\prod_{j}U_{j}^{m_{j}}$. We get

[TABLE]

as desired, having substituted $\eta=\xi-m$ and applied the result of Proposition 2.4. Now consider the general case $a=\sum_{m\in\mathbb{Z}^{n}}a_{m}\prod_{j}U_{j}^{m_{j}}$. Since $\alpha_{s}$ is an automorphism on $A_{\theta}$, we get

[TABLE]

and we are done. ∎

3 Asymptotic formula for the symbol of the adjoint

of a pseudo-differential operator

Here we prove the formula for the symbol of the adjoint for the noncommutative $n$ torus, adapting the proof of Lemma 1.2.3 of [12] to the noncommutative $n$ torus.

Theorem 3.1**.**

Suppose $P$ is a pseudodifferential operator with symbol $\sigma(P)=\rho=\rho(\xi)$ of order $M$. Then the symbol of the adjoint $P^{*}$ is of order $M$ and satisfies

[TABLE]

Proof.

Let $a,b\in A_{\theta}^{\infty}$. We have

[TABLE]

where

[TABLE]

so

[TABLE]

We have

[TABLE]

where

[TABLE]

so the corresponding symbol is

[TABLE]

where

[TABLE]

It remains to show that this symbol is of order $M$. Obviously,

[TABLE]

so we need to show that the remainder is of order $M-N_{1}$. Note that

[TABLE]

Integrating by parts, we get

[TABLE]

where, for arbitrary $a=\sum_{m}a_{m}\prod_{j}U_{j}^{m_{j}}$,

[TABLE]

Since $\rho\in S^{M}$ and $|\ell|=N_{1}$ we have $\delta^{\ell}\partial^{\ell}\rho\in S^{M-N_{1}}$. We get the boundedness of

[TABLE]

because $\partial^{\ell}R_{N_{1}}(y,\xi)$ is the rest of index $N_{1}$ in the Taylor expansion of $\partial^{\ell}(\xi-y\gamma)$ for which one has $\partial^{\ell}\rho\in S^{M-N_{1}}$. ∎

4 Asymptotic formula for the symbol of a product

of two pseudo-differential operators

Next we prove the formula for the product or composition of symbols for the noncommutative $n$ torus, adapting the proof of Theorem 7.1 of [14].

Theorem 4.1**.**

Suppose that $P$ is a pseudodifferential operator with symbol $\sigma(P)=\rho=\rho(\xi)$ of order $M_{1}$, and $Q$ is a pseudodifferential operator with symbol $\sigma(Q)=\phi=\phi(\xi)$ of order $M_{2}$. Then the symbol of the product $QP$ is of order $M_{1}+M_{2}$ and satisfies

[TABLE]

where $\partial^{\ell}:=\prod_{j}\partial_{j}^{\ell_{j}}$ and $\delta^{\ell}:=\prod_{j}\delta_{j}^{\ell_{j}}$.

Proof.

We want to show that if $\rho:\mathbb{R}^{n}\rightarrow A_{\theta}^{\infty}$ is of order $M_{1}$ and $\phi:\mathbb{R}^{n}\rightarrow A_{\theta}^{\infty}$ is of order $M_{2}$, $P_{\phi}\circ P_{\rho}=P_{\mu}$ where $\mu$ is of order $M_{1}+M_{2}$ and has asymptotic expansion

[TABLE]

Let $\{\varphi_{k}\}$ be the partition of unity constructed in Theorem 6.1 of [14] and define $\phi_{k}(\xi):=\phi(\xi)\varphi_{k}(\xi)$. We have

[TABLE]

Summing over $k$ from zero to infinity and applying Fubini’s Theorem, we get

[TABLE]

so

[TABLE]

and the convergence of the series is absolute and uniform for all $a\in A_{\theta}^{\infty}$. We want to compute the symbol of $P_{\phi}\circ P_{\rho}$, but issues with convergence of integrals make it so we need to compute the symbol of $P_{\phi_{k}}\circ P_{\rho}$. Let $a\in A_{\theta}^{\infty}$ be arbitrary. Applying $P_{\phi_{k}}\circ P_{\rho}$ we get

[TABLE]

where

[TABLE]

having done the changes of variables $(x,y)=(s,s+t)$ and $(\sigma,\tau)=(\xi-\eta,\eta)$ and applied Proposition 2.4. This suggests that

[TABLE]

where $\mu(\tau)=\sum_{k=0}^{\infty}\mu_{k}(\tau)$. We need to show that $\mu$ is a symbol in $S^{M_{1}+M_{2}}$ and has our desired asymptotic expression. Define $\mu_{k}$ by

[TABLE]

for all $\xi\in\mathbb{R}^{n}$. By Taylor’s formula with integral remainder given in Theorem 6.3 of [14], we get

[TABLE]

where

[TABLE]

for all $y,\xi\in\mathbb{R}^{2}$. Substituting back into our expression for $\mu_{k}(\xi)$ we get

[TABLE]

where

[TABLE]

Expressing $\rho(\xi)$ as $\rho(\xi)=\sum_{m}\rho_{m}(\xi)\prod_{j=1}^{n}U_{j}^{m_{j}}$, we see that

[TABLE]

so

[TABLE]

Let $\mu(\xi)=\sum_{k=0}^{\infty}\mu_{k}(\xi)$. It remains to show that $\mu$ is of order $M_{1}+M_{2}$. Obviously,

[TABLE]

so we just need to show that the remainder is of order $M_{1}+M_{2}-N_{1}$. Note that

[TABLE]

where

[TABLE]

and

[TABLE]

Integrating by parts, we get

[TABLE]

where for arbitrary $a=\sum_{m}a_{m}\prod_{j}U_{j}^{m_{j}}$ we have

[TABLE]

Since $|\ell|=N_{1}$, we have $\partial^{\ell}(\phi_{k})\in S^{M_{2}-N_{1}}$ and $\delta^{\ell}\rho\in S^{M_{1}}$. We get the boundedness of

[TABLE]

since $\delta^{\ell}\rho\in S^{M_{1}}$ and $\partial^{\ell}R_{N_{1}}(y,\xi)$ is the rest of index $N_{1}$ in Taylor’s expansion of $\partial^{\ell}\phi_{k}(\xi+\gamma y)$ for which on has $\partial^{\ell}\phi_{k}\in S^{M_{2}-N_{1}}$. ∎

5 Sobolev spaces on the noncommutative $n$ torus

Let $\lambda(\xi)=(1+\xi_{1}^{2}+\cdots+\xi_{n}^{2})^{1/2}$. Consider the following inner product on $A_{\theta}^{\infty}$.

Definition 5.1**.**

Define the Sobolev inner product $\langle\cdot,\cdot\rangle_{s}:A_{\theta}^{\infty}\times A_{\theta}^{\infty}\rightarrow\mathbb{C}$ by

[TABLE]

Note that for $s=0$ this agrees with $\langle\cdot,\cdot\rangle$. This inner product induces the following norm.

Definition 5.2**.**

Define the Sobolev norm $||\cdot||_{s}:A_{\theta}^{\infty}\rightarrow\mathbb{R}_{\geq 0}$ by

[TABLE]

Using this norm, we can define the analog of Sobolev spaces on the noncommutative $n$ torus.

Definition 5.3**.**

Define the Sobolev space $H^{s}$ to be the completion of $A_{\theta}^{\infty}$ with respect to $||\cdot||_{s}$.

We can prove that a pseudo-differential operator of order $d\in\mathbb{R}$ continuously maps $H^{s}$ into $H^{s-d}$. However we must first prove the case where $s=d$.

Theorem 5.4**.**

Suppose $\rho\in S^{d}$. Then $||P_{\rho}(a)||_{0}\leq C||a||_{d}$ for some constant $C>0$ and $P_{\rho}$ defines a bounded operator $P_{\rho}:H^{d}\rightarrow H^{0}$.

Proof.

Note that $\{\prod_{j}U_{j}^{m_{j}}:m\in\mathbb{Z}^{n}\}$ is an orthogonal basis with respect to $\langle\cdot,\cdot\rangle_{s}$ which is orthonormal in the case $s=0$. We have $||\prod_{j}U_{j}^{m_{j}}||_{0}^{2}=1$, $||\rho_{m}(\xi)\prod_{j}U_{j}^{m_{j}}||_{0}^{2}=|\rho_{m}(\xi)|^{2}$, and $||\rho(\xi)||_{0}^{2}=\sum_{m}|\rho_{m}(\xi)|^{2}$. Since $\rho$ is of order $d$, we have $||\rho(\xi)||_{0}\leq C_{\rho}(1+|\xi|)^{d}$, and since $(1-|\xi|)^{2}\geq 0$ gives us $(1+|\xi|)^{2}\leq 2(1+|\xi|^{2})$, we have

[TABLE]

Let $k_{\rho}:=C_{\rho}^{2}2^{d}$. Then we have

[TABLE]

Let $e_{s,m}:=(1+|m_{1}|^{2}+\cdots+|m_{n}|^{2})^{-s/2}\prod_{j}U_{j}^{m_{j}}$ and $E_{s}:=\{e_{s,m}\mid m\in\mathbb{Z}^{n}\}$. By definition we have $E_{s}$ orthonormal with respect to $\langle\cdot,\cdot\rangle_{s}$. It suffices to prove this theorem for the case $a=e_{d,m}$ by the orthonormality of $E_{d}$ since

[TABLE]

and

[TABLE]

Since $||e_{d,m}||_{d}^{2}=1$, it suffices to show that

[TABLE]

for some constant $K>0$. We have

[TABLE]

where

[TABLE]

so our desired constant is $K=k_{\rho}=C_{\rho}^{2}2^{d}$ and we are done. ∎

For the general case $s\neq d$ we need to prove a lemma saying that $||\cdot||_{s}=||\cdot||_{s-t}\circ P_{\lambda^{t}}$.

Lemma 5.5**.**

For any $a\in A_{\theta}^{\infty}$ and $s,t\in\mathbb{R}$, $a\in H^{s}$ if and only if $P_{\lambda^{t}}(a)\in H^{s-t}$ with $||a||_{s}=||P_{\lambda^{t}}(a)||_{s-t}$.

Proof.

Suppose that $a\in H^{s}$ or $P_{\lambda^{t}}(a)\in H^{s-t}$. Then

[TABLE]

so we know that $a\in H^{s}$ and $P_{\lambda^{t}}(a)\in H^{s-t}$. ∎

Then the general case follows quite easily.

Corollary 5.6**.**

Suppose $\rho\in S^{d}$. Then $||P_{\rho}(a)||_{s-d}\leq C||a||_{s}$ for some constant $C>0$ and $P_{\rho}$ defines a bounded operator $P_{\rho}:H^{s}\rightarrow H^{s-d}$.

Proof.

By Lemma 5.5, we have $||P_{\rho}(a)||_{s-d}=||P_{\lambda^{s-d}}(P_{\rho}(a))||_{0}$. By Theorem 4.1, the symbol $\sigma(P_{\lambda^{s-d}}\circ P_{\rho})$ is of order $d+(s-d)=s$, so Theorem 5.4 gives us $||P_{\lambda^{s-d}}(P_{\rho}(a))||_{0}\leq C||a||_{s}$ for some constant $C>0$. ∎

We can also define an analog of the $C^{k}$ norm for the noncommutative $n$ torus.

Definition 5.7**.**

Define the $C^{k}$ norm $||\cdot||_{\infty,k}:A_{\theta}^{\infty}\rightarrow\mathbb{R}_{\geq 0}$ as follows:

[TABLE]

where the $C^{*}$ norm $||\cdot||_{C^{*}}$ is given by

[TABLE]

Since, for arbitrary $a=\sum_{m}a_{m}\prod_{j}U_{j}^{m_{j}}$,

[TABLE]

we have $A_{\theta}^{k}=C^{k}$.

We can easily prove an analog of the Sobolev lemma as follows.

Theorem 5.8**.**

For $s>k+1$, $H^{s}\subseteq A_{\theta}^{k}$.

Proof.

First consider the case $k=0$. Note that $||\cdot||_{\infty,0}=||\cdot||_{C^{*}}$ so for arbitrary $a_{m}\prod_{j}U_{j}^{m_{j}}$ we have

[TABLE]

and for arbitrary $a=\sum_{m}a_{m}\prod_{j}U_{j}^{m_{j}}$ we have

[TABLE]

by the triangle inequality. We have

[TABLE]

so by the Cauchy-Schwarz inequality we get

[TABLE]

Since $2s>2$, $(1+|m_{1}|^{2}+\cdots+|m_{n}|^{2})^{-s}$ is summable over $m\in\mathbb{Z}^{n}$ and $||a||_{0}^{2}\leq C||a||_{s}$. Thus we get $||a||_{\infty,0}\leq||a||_{0}\leq C||a||_{s}$ and $H^{s}\subseteq A_{\theta}^{0}$.

Now suppose $k>0$. Using what we’ve proven for the previous case, we have

[TABLE]

for $|\ell|\leq k$ since $s-|\ell|\geq s-k>1$. Therefore,

[TABLE]

and we get $H^{s}\subseteq A_{\theta}^{k}$. ∎

We get the following corollary.

Corollary 5.9**.**

$\displaystyle\bigcap_{s\in\mathbb{R}}H^{s}=A_{\theta}^{\infty}$.

Proof.

Suppose $a\in\bigcap_{s\in\mathbb{R}}H^{s}$. Then for any $k\in\mathbb{Z}_{\geq 0}$, $a\in H^{k+2}$, so by the theorem we just proved, $a\in A_{\theta}^{k}$. Consequently $a\in A_{\theta}^{\infty}$, so $\bigcap_{s\in\mathbb{R}}H^{s}\subseteq A_{\theta}^{\infty}$.

Suppose $a\in A_{\theta}^{\infty}$. Then since $H^{s}$ is the completion of $A_{\theta}^{\infty}$ with respect to $||\cdot||_{s}$, $A_{\theta}^{\infty}\subseteq H^{s}$ for all $s\in\mathbb{R}$, and $A_{\theta}^{\infty}\subseteq\bigcap_{s\in\mathbb{R}}H^{s}$. ∎

We can also prove an analog of the Rellich lemma for the noncommutative $n$ torus.

Theorem 5.10**.**

Let $\{a_{N}\}\in A_{\theta}^{\infty}$ be a sequence. Suppose that there is a constant $C$ so that $||a_{N}||_{s}\leq C$ for all $N$. Let $s>t$. Then there is a subsequence $\{a_{N_{j}}\}$ that converges in $H^{t}$.

Proof.

Let $e_{s,m}:=(1+|m_{1}|^{2}+\cdots+|m_{n}|^{2})^{-s/2}\prod_{j=1}^{n}U_{j}^{m_{j}}$ and $E_{s}:=\{e_{s,m}\mid m\in\mathbb{Z}^{n}\}$. $E_{s}$ is an orthonormal basis with respect to $\langle\cdot,\cdot\rangle_{s}$, so we can write $a_{N}:=\sum_{k}a_{N,k}e_{s,k}$. Then

[TABLE]

and $|a_{N,k}|\leq C$. Applying the Arzela-Ascoli theorem to $\{a_{N,k}\}$ for some fixed $k$, we can get a subsequence $\{a_{N_{j},k}\}$ of $\{a_{N,k}\}$ such that for any $\epsilon>0$ there exists $M(\epsilon)\in\mathbb{N}$ such that $|a_{N_{i},k}-a_{N_{j},k}|<\epsilon$ whenever $i,j\geq M(\epsilon)$. Do this for all $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq r$, replacing $\{a_{N}\}$ with $\{a_{N_{j}}\}$ each time. Then we get a subsequence $\{a_{N_{j}}\}$ of $\{a_{N}\}$ such that for any $\epsilon>0$ there exists $M(\epsilon)\in\mathbb{N}$ such that, for all $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq r$, $|a_{N_{i},k}-a_{N_{j},k}|<\epsilon$ whenever $i,j\geq M(\epsilon)$. Now consider the sum

[TABLE]

Decompose it into two parts: one where $|k_{1}|^{2}+\cdots+|k_{n}|^{2}>r^{2}$ and one where $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq r$. On $|k_{1}|^{2}+\cdots+|k_{n}|^{2}>r^{2}$ we estimate

[TABLE]

so that

[TABLE]

If $\epsilon>0$ is given, we choose $r$ so that $2C^{2}(1+r^{2})^{t-s}<\epsilon$. The remaining part of the sum is over $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq r^{2}$ and can be bounded above by $\epsilon^{\prime}:=\epsilon-2C^{2}(1+r^{2})^{t-s}$ if $i,j\geq M(\sqrt[n]{\epsilon^{\prime}}/(2r+1))$ because a ball of radius $r$ centered at the origin is contained in a cube of side length $2r$ that has $(2r+1)^{n}$ lattice points. Then the total sum is bounded above by $\epsilon$, and we are done. ∎

6 The pseudodifferential calculus on finitely generated projective

modules over the noncommutative $n$ torus

We can generalize these results to arbitrary finitely generated projective right modules over the noncommutative $n$ torus following p. 553 of [4], which considers finitely generated projective modules over an arbitrary unital $*$-algebra. Let $E$ be a finitely generated projective right $A_{\theta}^{\infty}$-module. Since $E$ is a finitely generated projective right $A_{\theta}^{\infty}$-module, we can write $E$ as a direct summand $E=(A_{\theta}^{\infty})^{r}e$ of a free module $(A_{\theta}^{\infty})^{r}$ with direct complement $F=(A_{\theta}^{\infty})^{r}(\mathrm{id}-e)$, where the idempotent $e\in M_{r}(A_{\theta}^{\infty})$ is self-adjoint. Consider an $r\times r$ matrix valued symbol $\rho=(\rho_{j,k})$ where $\rho_{j,k}:\mathbb{R}^{n}\rightarrow A_{\theta}^{\infty}$ are scalar symbols and $\rho_{j,k}\in S^{d}$. Define the operator $P_{\rho}:E\rightarrow E$ as follows:

[TABLE]

Define the inner product $\langle\vec{a},\vec{b}\rangle:E\times E\rightarrow\mathbb{C}$ sending $(\vec{a},\vec{b})\mapsto\tau(\vec{b}^{*}\vec{a})$. Since

[TABLE]

Lemma 2.5 generalizes to $E$ as follows after applying it to each component:

[TABLE]

Theorems 3.1 and 4.1 generalize as follows.

Theorem 6.1**.**

1. (a)

For a pseudodifferential operator $P$ with $r\times r$ matrix valued symbol $\sigma(P)=\rho=\rho(\xi)$, the symbol of the adjoint $P^{*}$ satisfies

[TABLE] 2. (b)

If $Q$ is a pseudodifferential operator with $r\times r$ matrix valued symbol $\sigma(Q)=\rho^{\prime}=\rho^{\prime}(\xi)$, then the product $PQ$ is also a pseudodifferential operator and has symbol

[TABLE]

Proof.

First let’s prove part (a). Let $\rho$ be an $r\times r$ matrix valued symbol of order $M$ and $\vec{a},\vec{b}\in E$. We have

[TABLE]

where

[TABLE]

so

[TABLE]

The rest of the proof reduces to the $r=1$ case, applying it to each entry in $\rho=(\rho_{j,k})$.

We proceed to part (b). Let $\rho$ be an $r\times r$ matrix valued symbol of order $m_{1}$ and $\phi$ be an $r\times r$ matrix valued symbol of order $m_{2}$. Let $\{\varphi_{k}\}$ be a partition of unity and define $\phi_{k}(\xi):=\phi(\xi)\varphi_{k}(\xi)$. Let $\vec{a}\in E$. We have

[TABLE]

where

[TABLE]

so

[TABLE]

where $\lambda(\tau)=\sum_{k=0}^{\infty}\lambda_{k}(\tau)$.

Let

[TABLE]

Since

[TABLE]

the rest of the proof reduces to the $r=1$ case, applying it to each summand in the above sum. ∎

Let $\lambda(\xi)=(1+\xi_{1}^{2}+\cdots+\xi_{n}^{2})^{1/2}\mathrm{id}_{E}$. Consider the following inner product on $E$.

Definition 6.2**.**

Define the Sobolev inner product $\langle\cdot,\cdot\rangle_{s}:E\times E\rightarrow\mathbb{C}$ by

[TABLE]

Note that for $s=0$ this agrees with $\langle\cdot,\cdot\rangle$. This inner product induces the following norm.

Definition 6.3**.**

Define the Sobolev norm $||\cdot||_{s}:E\rightarrow\mathbb{R}_{\geq 0}$ by

[TABLE]

Using this norm, we can define the analog of Sobolev spaces on $E$.

Definition 6.4**.**

Define the Sobolev space $H^{s}$ to be the completion of $E$ with respect to $||\cdot||_{s}$.

We can prove that a pseudo-differential operator of order $d\in\mathbb{R}$ continuously maps $H^{s}$ into $H^{s-d}$. However we must first prove the case where $s=d$.

Theorem 6.5**.**

Suppose $\rho$ is a matrix valued symbol of order $d$. Then, for any $\vec{a}\in E$, $||P_{\rho}(\vec{a})||_{0}\leq C||\vec{a}||_{d}$ for some constant $C>0$ and $P_{\rho}$ defines a bounded operator $P_{\rho}:H^{d}\rightarrow H^{0}$.

Proof.

Let $F:=\{f_{j}:1\leq j\leq r\}$ be an orthogonal eigenbasis of $e$ normalized with respect to $\langle\cdot,\cdot\rangle$. Note that $\{\prod_{g}U_{g}^{m_{g}}f_{j}:m\in\mathbb{Z}^{n},1\leq j\leq r\}$ is an orthogonal basis of $E$ considered as a $\mathbb{C}$-vector space, with respect to $\langle\cdot,\cdot\rangle_{s}$. We have $||\prod_{g}U_{g}^{m_{g}}f_{j}||_{0}^{2}=1$, $||\rho_{m}(\xi)_{h,j}\prod_{g}U_{g}^{m_{g}}f_{j}||_{0}^{2}=|\rho_{m}(\xi)_{h,j}|^{2}$, and $||\rho(\xi)_{h,j}f_{j}||_{0}^{2}=\sum_{m}|\rho_{m}(\xi)_{h,j}|^{2}$. Since $\rho_{h,j}$ is of order $d$, we have $||\rho(\xi)_{h,j}||_{0}\leq C_{\rho}(1+|\xi|)^{d}$, and since $(1-|\xi|)^{2}\geq 0$ gives us $(1+|\xi|)^{2}\leq 2(1+|\xi|^{2})$, we have

[TABLE]

Let $k_{\rho}:=C_{\rho}^{2}2^{d}$. Then we have

[TABLE]

Let $e_{s,m}:=(1+|m_{1}|^{2}+\cdots+|m_{n}|^{2})^{-s/2}\prod_{g}U_{g}^{m_{g}}$ and $E_{s}:=\{e_{s,m}\mid n,m\in\mathbb{Z}\}$. By definition we have $E_{s}F$ orthonormal with respect to $\langle\cdot,\cdot\rangle_{s}$. It suffices to prove this theorem for the case $\vec{a}=e_{d,m}f_{j}$ by the orthonormality of $E_{d}F$ since

[TABLE]

and

[TABLE]

Since $||e_{d,m}f_{j}||_{d}^{2}=1$, it suffices to show that

[TABLE]

for some constant $K>0$. We have

[TABLE]

where

[TABLE]

so our desired constant is $K=rk_{\rho}=rC_{\rho}^{2}2^{d}$ and we are done. ∎

For the general case $s\neq d$ we need to prove a lemma saying that $||\cdot||_{s}=||\cdot||_{s-t}\circ P_{\lambda^{t}}$.

Lemma 6.6**.**

For any $\vec{a}\in E$ and $s,t\in\mathbb{R}$, $\vec{a}\in H^{s}$ if and only if $P_{\lambda^{t}}(\vec{a})\in H^{s-t}$ with $||\vec{a}||_{s}=||P_{\lambda^{t}}(\vec{a})||_{s-t}$.

Proof.

Suppose that $\vec{a}\in H^{s}$ or $P_{\lambda^{t}}(\vec{a})\in H^{s-t}$. Then

[TABLE]

so we know that $\vec{a}\in H^{s}$ and $P_{\lambda^{t}}(\vec{a})\in H^{s-t}$. ∎

Then the general case follows quite easily.

Corollary 6.7**.**

Suppose $\rho$ is a matrix valued symbol of order $d$. Then $||P_{\rho}(\vec{a})||_{s-d}\leq C||\vec{a}||_{s}$ for some constant $C>0$ and $P_{\rho}$ defines a bounded operator $P_{\rho}:H^{s}\rightarrow H^{s-d}$.

Proof.

By Lemma 6.6, we have $||P_{\rho}(\vec{a})||_{s-d}=||P_{\lambda^{s-d}}(P_{\rho}(\vec{a}))||_{0}$. By Proposition 6.1(b), the matrix valued symbol $\sigma(P_{\lambda^{s-d}}\circ P_{\rho})$ is of order $d+(s-d)=s$, so Theorem 6.5 gives us $||P_{\lambda^{s-d}}(P_{\rho}(\vec{a}))||_{0}\leq C||\vec{a}||_{s}$ for some constant $C>0$. ∎

We can also define an analog of the $C^{k}$ norm on $E$.

Definition 6.8**.**

Define the $C^{k}$ norm $||\cdot||_{\infty,k}:E\rightarrow\mathbb{R}_{\geq 0}$ as follows:

[TABLE]

where the $C^{*}$ norm $||\cdot||_{C^{*}}$ is given by

[TABLE]

Since, for arbitrary $\vec{a}=\sum_{j,m}a_{j,m}\prod_{g}U_{g}^{m_{g}}f_{j}$,

[TABLE]

we have $(A_{\theta}^{k})^{r}e=C^{k}$.

We can easily prove an analog of the Sobolev lemma on $E$ as follows.

Theorem 6.9**.**

For $s>k+1$, $H^{s}\subseteq(A_{\theta}^{k})^{r}e$.

Proof.

First consider the case $k=0$. Note that $||\cdot||_{\infty,0}=||\cdot||_{C^{*}}$ so for arbitrary $a_{j,m}\prod_{g=1}^{n}U_{g}^{m_{g}}f_{j}$ we have

[TABLE]

and for arbitrary $\vec{a}=\sum_{j,m}a_{j,m}\prod_{g=1}^{n}U_{g}^{m_{g}}f_{j}$ we have

[TABLE]

by the triangle inequality. We have

[TABLE]

so by the Cauchy-Schwarz inequality we get

[TABLE]

Since $2s>2$, $(1+|m_{1}|^{2}+\cdots+|m_{n}|^{2})^{-s}$ is summable over $m\in\mathbb{Z}^{n}$ and $j\in\{1,\ldots,r\}$ so $||\vec{a}||_{0}^{2}\leq C||\vec{a}||_{s}$. Thus we get $||\vec{a}||_{\infty,0}\leq||\vec{a}||_{0}\leq C||\vec{a}||_{s}$ and $H^{s}\subseteq(A_{\theta}^{0})^{r}e$.

Now suppose $k>0$. Using what we’ve proven for the previous case, we have

[TABLE]

for $|\ell|\leq k$ since $s-|\ell|\geq s-k>1$. Therefore,

[TABLE]

and we get $H^{s}\subseteq(A_{\theta}^{k})^{r}e$. ∎

We get the following corollary.

Corollary 6.10**.**

$\displaystyle\bigcap_{s\in\mathbb{R}}H^{s}=(A_{\theta}^{\infty})^{r}e$.

Proof.

Suppose $a\in\bigcap_{s\in\mathbb{R}}H^{s}$. Then for any $k\in\mathbb{Z}_{\geq 0}$, $\vec{a}\in H^{k+2}$, so by the theorem we just proved, $\vec{a}\in(A_{\theta}^{k})^{r}e$. Consequently $\vec{a}\in(A_{\theta}^{\infty})^{r}e$, so $\bigcap_{s\in\mathbb{R}}H^{s}\subseteq(A_{\theta}^{\infty})^{r}e$.

Suppose $a\in(A_{\theta}^{\infty})^{r}e$. Then since $H^{s}$ is the completion of $(A_{\theta}^{\infty})^{r}e$ with respect to $||\cdot||_{s}$, $(A_{\theta}^{\infty})^{r}e\subseteq H^{s}$ for all $s\in\mathbb{R}$, and $(A_{\theta}^{\infty})^{r}e\subseteq\bigcap_{s\in\mathbb{R}}H^{s}$. ∎

We can also prove an analog of the Rellich lemma on $E$.

Theorem 6.11**.**

Let $\{\vec{a}_{N}\}\in(A_{\theta}^{\infty})^{r}e$ be a sequence. Suppose that there is a constant $C$ so that $||\vec{a}_{N}||_{s}\leq C$ for all $N$. Let $s>t$. Then there is a subsequence $\{\vec{a}_{N_{j}}\}$ that converges in $H^{t}$.

Proof.

Let $F:=\{f_{j}:1\leq j\leq r\}$ be a set of eigenvectors of $e$ normalized with respect to $\langle\cdot,\cdot\rangle$, where $f_{j}$ has corresponding eigenvalue $\lambda_{j}$ for $1\leq j\leq r$. Let $e_{s,m}:=(1+|m_{1}|^{2}+\cdots+|m_{n}|^{2})^{-s/2}\prod_{g}U_{g}^{m_{g}}$ and $E_{s}:=\{e_{s,m}\mid n,m\in\mathbb{Z}\}$. $E_{s}F$ is an orthonormal basis with respect to $\langle\cdot,\cdot\rangle_{s}$, so we can write $\vec{a}_{N}:=\sum_{h,k}a_{N,h,k}e_{s,k}f_{h}$. Then

[TABLE]

and $|a_{N,h,k}|\leq C$. Applying the Arzela-Ascoli theorem to $\{a_{N,h,k}\}$ for some fixed $(h,k)$, we can get a subsequence $\{a_{N_{j},h,k}\}$ of $\{a_{N,h,k}\}$ such that for any $\epsilon>0$ there exists $M(\epsilon)\in\mathbb{N}$ such that $|a_{N_{i},h,k}-a_{N_{j},h,k}|<\epsilon$ whenever $i,j\geq M(\epsilon)$. Do this for all $1\leq h\leq r$ and $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq R$, replacing $\{a_{N}\}$ with $\{a_{N_{j}}\}$ each time. Then we get a subsequence $\{a_{N_{j}}\}$ of $\{a_{N}\}$ such that for any $\epsilon>0$ there exists $M(\epsilon)\in\mathbb{N}$ such that, for all $1\leq h\leq r$ and $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq R$, $|a_{N_{i},h,k}-a_{N_{j},h,k}|<\epsilon$ whenever $i,j\geq M(\epsilon)$. Now consider the sum

[TABLE]

Decompose it into two parts: one where $|k_{1}|^{2}+\cdots+|k_{n}|^{2}>R^{2}$ and one where $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq R^{2}$. On $|k_{1}|^{2}+\cdots+|k_{n}|^{2}>R^{2}$ we estimate

[TABLE]

so that

[TABLE]

If $\epsilon>0$ is given, we choose $R$ so that $2rC^{2}(1+R^{2})^{t-s}<\epsilon$. The remaining part of the sum is over $|k_{1}|^{2}+\cdots+|k_{n}|^{2}\leq R^{2}$ and can be bounded above by $\epsilon^{\prime}:=\epsilon-2rC^{2}(1+R^{2})^{t-s}$ if $i,j\geq M(\sqrt[n]{\epsilon^{\prime}}/(2R+1))$ because a ball of radius $R$ centered at the origin is contained in a cube of side length $2R$ that has $(2R+1)^{2}$ lattice points. Then the total sum is bounded above by $\epsilon$, and we are done. ∎

7 Acknowledgement

I would like to thank Farzad Fathizadeh for suggesting the problem of filling in the details in the pseudodifferential calculus on the noncommutative $n$ torus. I would also like thank Matilde Marcolli for helping me make plans to take my candidacy exam. I would like to thank Vlad Markovic and Eric Rains for agreeing to be on my candidacy exam committee.